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PHYSICAL REVIEW D, VOLUME 59, 074020Vector meson dominance and ther meson
M. Benayoun*LPNHE des Universites Paris VI et VIIIN2P3, Paris, France
H. B. OConnell
Department of Physics and Astronomy, University of Kentucky, Lexington, Kentucky 405060055
A. G. Williams
Department of Physics and Mathematical Physics and Special Research Centre for the Subatomic Structure of Matter,University of Adelaide 5005, Australia
~Received 29 July 1998; published 8 March 1999!
We discuss the properties of vector mesons, in particular ther0, in the context of the hidden local symmetry~HLS! model. This provides a unified framework to study several aspects of the low energy QCD sector. First,we show that in the HLS model the physical photon is massless, without requiring off field diagonalization. Wethen demonstrate the equivalence of HLS and the two existing representations of vector meson dominance,VMD1 and VMD2, at both the tree level and one loop order. Finally theS matrix pole position is shown toprovide a model and process independent means of specifying ther mass and width, in contrast with the realaxis prescription currently used in the Particle Data Group tables.@S05562821~99!028076#
PACS number~s!: 12.40.Vv, 11.30.Qc, 11.30.Rd, 12.39.Fewth
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I. INTRODUCTION
There is no reliable analytic means for calculating loand medium energy strongly interacting processes withunderlying theory, QCD. Despite progress in numerical sties of QCD both via the lattice@1# and QCDmotivated models @2#, preQCD effective Lagrangians involving the oserved hadronic spectrum continue to play an importantin studies of this sector. We shall be concerned in this wwith the interactions of the pseudoscalar and vector mesas described by the hidden local symmetry~HLS! model@3#.
The focus of our paper will be ther resonance. As thelightest and broadest of the vector octets it plays an imptant role in phenomenology and is presently the subjecinterest as a possible indicator of chiral symmetry restorain heavy ion collisions@4#. It also serves as a guide for phyics in other sectors. As we shall see the interaction of vemesons and photons in the HLS construction is closanalogous to the SU~2!^U~1! symmetry breaking of theelectroweak interaction. The traditional determination ofr mass and width has been plagued by model dependewhich as we show can be avoided by use of theSmatrixpole, closely following developments in the study of theZ0.
Our paper is structured as follows: we begin with a broutline of the HLS model and discuss the generation of vtor boson masses by the spontaneous breaking of the gchiral symmetry through the HiggsKibble~HK! mechanism@5#. In Sec. III we consider the relationship between theHK masses and thephysical vector masses, using a twochannel propagator matrix for the photonr system. This allows one to consider the effect of mixing in the dressingthe bare propagators and when this is properly considered
*Email address: benayoun@in2p3.frEmail address: hoc@pa.uky.eduEmail address: awilliam@physics.adelaide.edu.au05562821/99/59~7!/074020~11!/$15.00 59 0740e
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dressed photon is seen to be massless as required, withe need for a change of basis.
Section IV is devoted to a comparison of the two comonly used representations of vector meson domina~VMD ! ~referred to hereafter as VMD1 and VMD2 following the convention of Ref.@6#!, both of which can be obtained as special cases from the HLS Lagrangian of Ref.@3#.Note that VMD2 is the most commonly used version andoften in the literature, simply referred to as VMD or thvector dominance model@7#. Using the pion and kaon formfactors we explicitly demonstrate their equivalence at the tlevel ~which is trivial! and at oneloop order, where carerequired. This treatment is performed in the general casa2, wherea is the HLS parameter@3#, for which we introduce VMD1a and VMD2a in an obvious manner.
Finally, as the vector mesons are resonant particles,study the effect of the large width on the determinationmodel independentr parameters. TheSmatrix pole positionis shown to provide a truly modelindependent and, furthmore, processindependent parametrization of ther meson.
II. HIDDEN LOCAL SYMMETRY AND VMD
The HLS model allows us to produce a theory with vecmesons as the gauge bosons of a hidden local symmThese then become massive because of the spontanbreaking of a chiralU(3)L ^ U(3)R global symmetry. Let usconsider the chiral Lagrangian@8#
Lchiral51
4Tr @]mF]
mF#, ~1!
whereF(x)5 f pU(x) in the usual notation. This exhibits thchiral U(3)L ^ U(3)R symmetry underUgLUgR . We canwrite this in exponential form and expand
F~x!5 f pe2iP~x!/ f p5 f p12iP~x!22P
2~x!/ f p1 ;~2!1999 The American Physical Society201
t aing,n by thes
M. BENAYOUN, H. B. OCONNELL, AND A. G. WILLIAMS PHYSICAL REVIEW D 59 074020therefore, substituting into Eq.~1! we see that the vacuum corresponds toP50, U51. That is,F has a nonzero vacuumexpectation value which spontaneously breaks theU(3)L ^ U(3)R symmetry as this symmetry of the Lagrangian is nosymmetry of the vacuum. The massless Goldstone bosons contained inP, associated with the spontaneous symmetry breakthen correspond to the perturbations about the QCD vacuum and we can think of expansions in this field as giveHermitian matrixP5PaTa where the GellMann matrices are normalized such that Tr@TaTb#5dab/2. So for the pseudoscalarone has
P51
& S 1& p01 1A6 p81 1) h0 p1 K1p2 2 1& p01 1A6 p81 1) h0 K0K2 K0 2A2
3p81
1
)h0
D . ~3!rge
t
tisde

try,he
a
e for
eenHowever, in addition to the global chiral symmetry,G,the HLS scheme includes a local symmetry,H, in Eq. ~1! inthe following way@3#. Let
U~x![jL~x!jR~x! ~4!
where
jR,L~x!5exp@ iS~x!/ f S#exp@6 iP~x!/ f p# ~5!
jR,L~x!h~x!jR,L~x!gR,L . ~6!
Note that this introduces the scalar field,S(x), analogous tothe pseudoscalar,P(x), of Eq. ~3!, though S(x) does notappear in the chiral fieldU(x) of Eq. ~4!. The general formsof the transformations are given bygL,R5exp(ieL,R
a Ta) andh(x)5exp@ieH
a (x)Ta#.One now seeks to incorporate HLS into the low ene
Lagrangian in a nontrivial way, thereby introducing thlightest vector meson states@3,9#. The procedure is to firsrewriteLchiral explicitly in terms of thej components:
Lchiral52f p
2
4Tr@]mjLjL
2]mjRjR #2. ~7!
The Lagrangian can be gauged for both electromagneand the hidden local symmetry by changing to covariantrivatives
DmjL,R5]mjL,R2 igVmjL,R1 iejL,RAmQ ~8!
whereAm is the photon field,Q5diag(2/3,21/3,21/3) andVm5Vm
a Ta whereVma are the vector meson fields transform
ing as Vmh(x)Vmh(x)1 ih(x)]mh(x)/g. Suppressingfor brevity the spacetime indexm, we can write the vectormeson field matrixVm as
V51
& S ~r01v!/& r1 K* 1r2 ~2r01v!/& K* 0K* 2 K* 0 f
D . ~9!
07402y
m
Here we have used ideal mixing in defining the barev andfmesons. Note that the vector meson fieldsVm
a 5K* ,r,v,fare introduced in the role of gauge fields forH[flavorSU~3!. However,LA[Lchiral is independent ofV, and in theHLS model a second piece of the Lagrangian,LV , is introduced:
LA52f p
2
4Tr@~DmjLjL
2DmjRjR !#2
LV52f p
2
4Tr@~DmjLjL
1DmjRjR !#2. ~10!
The full HLS Lagrangian is then finally defined by@3#
LHLS5LA1aLV , ~11!
where we see that the HLS parametera has now been introduced.
In the absence of the gauge fields,Vm andAm , we see thatEq. ~10! reduces to
LA51
2Tr@]mP]
mP#, LV5f p
2
2 f S2 Tr@]mS]
mS# ~12!
to quadratic order in bosons. In this caseP and S, wouldboth be Goldstone bosons, whereP(x) is associated with thespontaneous breaking of the usual, global chiral symmeG, of LA arising from the vacuum expectation value of tU(x) fields and analogously forS(x). However, gauge invariance allows for their elimination. It is usual to takespecial gauge, the unitary gauge, forH, for which the scalarfield no longer appears,S(x)50 @9# ~for a discussion of theunitary gauge and spontaneously broken symmetries, seexample Sec. 1253 of Ref.@10#!. This is phenomenologically reasonable as no chiral partner for the pion has bobserved. With this choice we have
jL~x!5jR~x![j~x!5exp@ iP~x!/ f p#. ~13!02
g
nl
ra
arof
s
VECTOR MESON DOMINANCE AND THEr MESON PHYSICAL REVIEW D59 074020By demandingS(x)50 the local symmetry,H, is lost, buttheg transformations of the global chiral symmetry groupGwill regenerate the scalar field through
j~x!j8~x!5j~x!g
5exp@ iS8P~x!,g/ f S#exp@ iP8~x!/ f p#. ~14!
However, the system can still maintain the unitarity gauconditionS(x)50 and global chiral symmetry,G, under thecombined transformation@9#
j~x!j8~x!5hP~x!,gj~x!g,
hP~x!,g5exp@2 iS8P~x!,g/ f S# ~15!
where the particular choice of local transformatiohP(x),g, kills the scalar field created by the globa07402e
,
transformationg. The physical meaning of this is that vectofields acquire longitudinal components by eating the sclar S field through a transformation of the form~to lowestorder in Goldstone fields!
VmVm21
g fS]mS8. ~16!
Once the scalar field is effectively removed by the unitity gauge choice the traditional VMD Lagrangian, thatVMD2, is obtained from an expansion ofLHLS in the pionfield, as per Eq.~2!. The HLS model actually generalizeVMD2, through the additional parameter,a of Eq. ~11!; sowe shall refer to the resulting Lagrangian as VMD2a . It hasthe formLVMD2a521
4FmnF
mn21
4VmnV
mn2ae fp2 gFr1 v3 2&3 f GA
12
3ae2f p
2 A21a fp
2 g2
2~r21v21f2!1a fp
2 g2~r1r21K* 1K* 21K* 0K* 0!
1]p1]p21]K1]K21]K0]K011
2@]p0]p01]p8]p81]h0]h0#
1i
2@agr1e~22a!A#@]p1p22]p2p1#
1i
4@ag~r1v2&f!12e~22a!A#~]K1K22]K2K1!
1iag
4@r2v1&f#@]K0K02]K0K0#1
iag
2&r1@]K0K22]K2K01&~]p2p02]p0p2!#
1iag
2&r2@]K1K02]K0K11&~]p0p12]p1p0!#
1iag
4K* 0@]p0K02]K0p01&~]K2p12]p1K2!1)~]K0p82]p8K0!#
1iag
4K* 0@]K0p02]p0K01&~]p2K12]K1p2!1)~]p8K02]K0p8!#
1iag
4K* 2@]p0K12]K1p01&~]p1K02]K0p1!1)~]p8K12]K1p8!#
1iag
4K* 1@]K2p02]p0K21&~]K0p22]p2K0!1)~]K2p82]p8K2!#, ~17!03
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M. BENAYOUN, H. B. OCONNELL, AND A. G. WILLIAMS PHYSICAL REVIEW D 59 074020where for brevity we used the notationr[r0. HereFmn andVmn are the standard Abelian and nonAbelian field strentensors for the photon and vector meson octet respectivThe photon and vector mesons therefore acquire Lagranmasses through the HK mechanism@5# with the vector mesons also obtaining longitudinal components through@16#. The HK mass generated for the vector mesons is giby
mHK2 [M25ag2f p
2 . ~18!
The choicea52 @3# reduces VMD2a @Eq. ~17!# to the traditional VMD2, eliminating the photonpion contact term anreproducing the KawarabayashiSuzukiRiazuddFayyazuddin~KSRF! relation @11# through Eq.~18!.
For simplicity we shall not consider SU~3! breaking dueto strangeness in the vector meson sector@12,13#, which results in changes in the vector meson coupling constantsHK masses for theK* andf. One should also note that thVMD2a Lagrangian does not generate couplings of the vtors to thesingletcomponent of the pseudoscalar mesons.be perfectly clear, VMD2a is nothing but the standard HLSLagrangian used to deduce the HLS pion form factor in R@14#. It is a nontrivial extension of the traditional vectomeson dominance model VMD2, sincea2 generates noonly a violation of the KSRF relation (mr
252g2f p2 ), but also
introduces a direct coupling of the photon to the pseudoslars.
III. DRESSED VECTOR PROPAGATORS AND THEPHOTON POLE
The physical photon is massless, but the VMD2a Lagrangian in Eq.~17! possesses a photon mass term. To fithe physical results one obtains from VMD2a , we need todress the vector propagators by means of a matrix equawhich accounts for the possible mixings between vector pticles @15#. The matrix DysonSchwinger equation is giveby
iD mn5 iD mn0 1 iD ma
0 iGabiD bn ~19!
whereDmn is the dressed propagator matrix andDmn0 is the
bare propagator obtained from the Lagrangian,
Dmn0 5S 2gmn1 qmqnM2 D 1q22M2 . ~20!
Inverting Eq.~19! we obtain
Dmn~21!5D0mn
~21!1Gmn . ~21!
The selfenergy matrixGmn is composed of two entitiesThe first is the polarization functionPmn generated fromloop effects. As can be shown from Eq.~17!, the vectormesons couple to loops only through conserved curre@13#, and hence the polarization tensor is transverse
Pmn[~gmn2qmqn /q2!P~q2! ~22!
and from the node theorem@15#07402hly.an
.n

nd
co
f.
a
d
onr
ts
P~0!50. ~23!
The functionP(s) has a branch cut along the real axis bginning at the production threshold~for the r this is s54mp
2 ) and extending to infinity. The second dressing tecomes from the Lagrangian mixing terms between the vemesons and photon. For simplicity let us consider the cwith only the photon and ther[r0 @3#; we then have
Gmn5S gmn2 qmqnq2 D S Pgg PgrPrg Prr D 1gmn eM2
g S 0 11 0D ,~24!
whereM is the HK mass of ther given in Eq.~18!.As we shall here only consider amplitudes such
e1e2p1p2 where the vectors couple to external coserved currents~lepton or pion! the qmqn pieces of Eq.~21!can be ignored. So defining the scalar part of the propagthrough Dmn(q)[gmnD(q
2) the surviving part of thedressed propagator is given by
S Dgg DgrDrg DrrD 5S e2M2/g21Pgg2s eM2/g1PgreM2/g1Prg M21Prr2sD 21,~25!
where s[q2. A similar formalism has been discussed bHung and Sakurai for theg2Z0 system in the electroweainteraction@16#. The poles,pi , i 51,2, of the two channepropagator are obtained from
Det@D21~pi !#50. ~26!
The VMD2a Lagrangian given in Eq.~17! is O(e2), wheree[e/g; so we work to this order in the calculation of thpole positions in the complexs plane. Equation~26! gives
2p5M2~11e2!1Prr1Pgg6H M2~11e2!1Prr2Pgg1
2e2~Pgg2Prr!M212eM2Pgr2Pgr
2
M2~11e2!1Prr2Pgg1O~e3!J .
~27!
In power counting with respect toPrr , we see thatPgr andPgg are intrinsically ofO~e! andO(e2) respectively: so wecan further simplify:
2p5M2~11e2!1Prr1Pgg6H M2~11e2!1Prr2Pgg2
2~e2M2Prr22eM2Pgr1Pgr
2 !
M21Prr1O~e3!J . ~28!
Thus the poles are, toO(e2),
pg5Pgg~pg!1e2M2Prr~pg!22eM
2Pgr~pg!1Pgr2 ~pg!
M21Prr~pg!~29!04
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VECTOR MESON DOMINANCE AND THEr MESON PHYSICAL REVIEW D59 074020pr5M2~11e2!1Prr~pr!
2e2M2Prr~pr!22eM
2Pgr~pr!1Pgr2 ~pr!
M21Prr~pr!.
~30!
Since they couple only through conserved currents, thelarization functions,P(0)50 @15#, and we see that the photon pole resides atq250 as required. The explicit photomass term in Eq.~17! is canceled by the photon selfenergSimilarly ther pole is shifted only by corrections ofO(e2)by mixing with the photon. This result is as we might expesince the vacuum stateF05 f p in invariant under the U~1!EM transformation@17#
FF1 i e~x!@Q,F#, ~31!
and massive vectors correspond to spontaneously brosymmetries. However, it is slightly more subtle than tusual case where invariance under a transformation ofform F0F01 i ea(x)TaF0 requires that there be no Lagrangian mass term~asTaF050). The generation of vectomeson masses in the presence of a massless photon is trin the general case by Gottlieb@18#.
HLS can therefore be used to demonstrate from...